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15 光的偏振

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15.1 线偏振

平行于槽的电场分量（E-field component parallel to the slots）：这部分电场会被吸收或反射。

垂直于槽的电场分量（E-field component perpendicular to the slots）：这部分电场会透过金属板。

This set of two linear polarizers produces LP light. What is the final intensity?

First LP transmits 1/2 of the unpolarized light:

\[ I_1 =\frac{1}{2} I_0 \]

Second LP projects out the \(E\)-field component parallel to the TA:

\[ \vec{E}_2 = (\vec{E}_1 \bullet \hat{n}) \hat{n} \implies E_2 = E_1 \cos \theta \]

又因为 \(I \propto E^2\)，我们可以得到

\[ I_2 = I_1 \cos^2 \theta \]

This result is called the Law of Malus (马隆定律) (for LP light incident on LP)

Question

15.2 其他偏振态

\[ \begin{aligned} E_x=E_{x_0}\sin(kz-\omega t+\phi_x)\\ E_y=E_{y_0}\sin(kz-\omega t+\phi_y)\\ \end{aligned} \]

我们可以认为线偏振的情况为

\[ \left\{ \begin{aligned} &\phi=\phi_x-\phi_y=0\\ &E_{y_0}/E_{x_0}=\tan \theta \end{aligned} \right. \]

那么圆偏振的情况为

\[ \left\{ \begin{aligned} &\phi=\phi_x-\phi_y=\pm \frac{\pi}2\\ &E_{y_0}=E_{x_0} \end{aligned} \right. \]

根据旋转方向的不同，可以分为左旋与右旋光

如果你把圆偏振光照射到吸收体上，原则上它会开始旋转→角动量守恒！

五种偏振类型

(1) Unpolarized Light (无偏振光, 自然光)

  • Random phase difference

(2) Linearly Polarized Light (线偏振光)

(3) Partial Polarized Light (部分偏振光)

(4) Circular Polarized Light (圆偏振光)

\[ \begin{aligned} &E_x = E_0 \sin(kz - \omega t + \frac{\pi}{2})\\ &E_y = E_0 \sin(kz - \omega t)\\ \end{aligned} \]

(5) Ellipse Polarized Light (椭圆偏振光)

\[ \begin{aligned} &E_x = E_1 \sin(kz - \omega t + \delta)\\ &E_y = E_2 \sin(kz - \omega t)\\ &\text{and } E_1 \neq E_2, \quad \text{or} \quad \delta \neq \pm \frac{\pi}{2} \end{aligned} \]

定义偏振度：

\[ P = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} \]

其中 \(I_{max}\) 和 \(I_{min}\) 分别为最大偏振方向和最小偏振方向的光强

Linearly Polarized Light: \(I_{\min} = 0, \quad P = 1\)

Unpolarized Light: \(I_{\max} = I_{\min}, \quad P = 0\)

15.3 Polarization by Reflection

$$ \begin{cases} \theta_p + \theta_r = 90^\circ \ n_1 \sin \theta_p = n_2 \sin \theta_r \end{cases} \Rightarrow n_1 \sin \theta_p = n_2 \sin(90^\circ - \theta_p) = n_2 \cos \theta_p

$$

\[ \therefore \tan \theta_p = \frac{n_2}{n_1} \quad \text{------------------ Brewster's Law} \]

当外界为空气的时候，\(n_1=1\)，那么这时 \(\tan \theta_p=n\)

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